What is algebraic function theory? What is algebraic function theory? In other words, what is an algebraic function field? 
I need to understand the gist of it. Could someone explain what this is? I am writing a paper on the beginnings of ring theory.
 A: An algebraic function field is the pair consisting of a field $k$ (not assumed algebraically closed) and and a finitely generated extension field $K$ of transcendence degree one:  $k\subset K=k(x_1,...,x_r), \; \operatorname {trdeg}_k K=1$.   
There is an equivalence of categories between smooth irreducible projective curves $C$ over $k$  and function fields $K$ over $k$, in which $K$ corresponds to the field of rational functions on $C$.  
Nowadays theoretical algebraic geometers tend to prefer the language of curves seen as schemes, whereas more applied mathematicians (and computer scientists) interested in the  theory of codes and in cryptography seem to like function fields more.
It is well known that Grothendieck didn't like valuations, the core tool in function fields, and the dichotomy I evoke might partly be due to the tremendous influence of his and Dieudonné's treatise EGA.   
A: I've never heard the term, "algebraic function theory." I've heard of algebraic functions; a function $f(x)$ is algebraic if there is a polynomial $p(t)$ such that $p(f(x))$ is identically zero. Is that what you are talking about?
